# nLab monoidal model category

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

for stable $(\infty,1)$-categories

for $(\infty,1)$-operads

for $(n,r)$-categories

for $(\infty,1)$-sheaves / $\infty$-stacks

#### Monoidal categories

monoidal categories

# Contents

## Idea

A monoidal model category is a model category which is also a closed monoidal category in a compatible way. In particular, its homotopy category inherits a closed monoidal structure, as does the (infinity,1)-category that it presents.

## Definition

###### Definition

A (symmetric) monoidal model category is a model category $\mathcal{C}$ equipped with the structure of a closed symmetric monoidal category $(\mathcal{C}, \otimes, I)$ such that the following two compatibility conditions are satisfied

1. (pushout-product axiom) For every pair of cofibrations $f \colon X \to Y$ and $f' \colon X' \to Y'$, their pushout-product, hence the induced morphism out of the cofibered coproduct over ways of forming the tensor product of these objects

$(X \otimes Y') \coprod_{X \otimes X'} (Y \otimes X') \longrightarrow Y \otimes Y' \,,$

is itself a cofibration, which, furthermore, is acyclic (i.e. a weak equivalence) if $f$ or $f'$ is.

(Equivalently this says that the tensor product $\otimes \colon C \times C \to C$ is a left Quillen bifunctor.)

2. (unit axiom) For every cofibrant object $X$ and every cofibrant resolution $\emptyset \hookrightarrow Q I \stackrel{p_I}{\longrightarrow} I$ of the tensor unit $I$, the resulting morphism

$Q I \otimes X \stackrel{p_I \otimes X}{\longrightarrow} I\otimes X \stackrel{\simeq}{\longrightarrow} X$

is a weak equivalence.

###### Remark

Let $X$ be a cofibrant object, hence $\varnothing \overset{\exists !}{\to} X$ a cofibration.

In this case the pushout-product axiom (Def. ) says that the tensor product functor $X \otimes (-)$ preserves cofibrations and acyclic cofibrations. Since the ambient category is assumed to be closed monoidal category, so that this functor has a right adjoint internal hom $[X,-]$, this means that it is the left Quillen functor in a Quillen adjunction

$\mathcal{C} \underoverset {\underset{[X,-]}{\longrightarrow}} {\overset{X\otimes(-)}{\longleftarrow}} { \;\;\;\;\; \bot_{\mathrlap{Qu}} \;\;\;\;\; } \mathcal{C}$

###### Remark

(cofibrant tensor unit implies unit axiom)
As a special case of Rem. : If the tensor unit $I$ happens to be cofibrant, then the unit axiom in def. is already implied by the pushout-product axiom.

(Because then $Q I \to I$ is a weak equivalence between cofibrant objects and such are preserved by functors that preserve acyclic cofibrations, by Ken Brown's lemma.)

###### Definition

(monoid axiom)

One says that a monoidal model category (Def. ) satisfies the monoid axiom if every morphism that is obtained as a transfinite composition of pushouts of tensor products $X\otimes f$ of acyclic cofibrations $f$ with any object $X$ is a weak equivalence.

###### Remark

In particular, the axiom in def. says that for every object $X$ the functor $X \otimes (-)$ sends acyclic cofibrations to weak equivalences.

## Properties

### Monoidal homotopy category

#### Abstractly

###### Proposition

For $(\mathcal{C},\otimes)$ a monoidal model category, def. , then the derived functor $\otimes^L$ of the tensor product makes the homotopy category of the model category itself into a monoidal category, such that the localization functor

$\gamma \;\colon\; (\mathcal{C},\otimes) \longrightarrow (Ho(\mathcal{C}), \otimes^L)$

is a lax monoidal functor.

###### Proof

Let $V$ be a monoidal model category, and consider it as a derivable category in the sense of (Shulman 11, section 8) with $V_Q$ the subcategory of cofibrant objects and $V_R=V$. Then $\otimes :V\times V\to V$ is left derivable, i.e. it preserves the $Q$-subcategories and weak equivalences. Since deriving is pseudofunctorial (and product-preserving) on the 2-category of derivable categories and left derivable functors, it follows immediately that $Ho(V)$ is monoidal; this is (Shulman 11, example 8.13).

Now let $V_0$ denote the category $V$ with its trivial derivable structure: only isomorphisms are weak equivalences, and all objects are both $Q$ and $R$. Then of course $Ho(V_0) = V$, and $V_0$ is also a pseudomonoid in derivable categories. The identity functor $Id : V_0 \to V$ is not left derivable, since it does not preserve $Q$-objects; but it is right derivable, since we took all objects in $V$ to be $R$-objects (ignoring the fibrant objects in the model structure on $V$). Of course $Id$ is strong monoidal, and this monoidality constraint can be expressed as a square in the double category of (Shulman 11) whose vertical arrows are left derivable functors and whose horizontal arrows are right derivable functors; moreover the axioms on a monoidal functor may be expressed using products and double-categorical pasting in this double category. Therefore, it is all preserved by the double pseudofunctor $Ho$; but $Ho(Id) = \gamma : V \to Ho(V)$. The only thing that is not visible to the double category is the invertibility of the monoidal constraint, and hence this is not preserved by the double pseudofunctor; thus $\gamma$ is only lax monoidal.

#### Explicitly

###### Proposition

Let $(\mathcal{C}, \otimes, 1)$ be a monoidal model category with cofibrant tensor unit. Then the left derived functor $\otimes^L$ of the tensor product exists

$\array{ \mathcal{C}_c\times \mathcal{C}_c &\overset{\otimes}{\longrightarrow}& \mathcal{C}_c \\ {}^{\mathllap{\gamma \times \gamma}}\downarrow &\swArrow_{\mu^{-1}}& \downarrow^{\mathrlap{\gamma}} \\ Ho(\mathcal{C})\times Ho(\mathcal{C}) &\overset{\otimes^L}{\longrightarrow}& Ho(\mathcal{C}) }$

and makes the homotopy category into a monoidal category $(Ho(\mathcal{C}), \otimes^L, \gamma(1))$.

Moreover, the localization functor

$\gamma \;\colon\; (\mathcal{C}_c, \otimes, 1) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(1))$

on the category of cofibrant objects is a strong monoidal functor with structure morphism the inverse of the above natural isomorpmism

$\mu_{X,Y} \;\colon\; \gamma(X)\otimes^L \gamma(Y) \longrightarrow \gamma(X \otimes Y) \,.$
###### Proof

For the left derived functor (def.) of the tensor product

$\otimes \; \mathcal{C}\times \mathcal{C} \longrightarrow \mathcal{C}$

to exist, it is sufficient that its restriction to the subcategory

$(\mathcal{C} \times \mathcal{C})_c \simeq \mathcal{C}_c \times \mathcal{C}_c$

of cofibrant objects preserves acyclic cofibrations (Ken Brown's lemma, here).

Every morphism $(f,g)$ in the product category $\mathcal{C}_{c}\times \mathcal{C}_{c}$ may be written as a composite of a pairing with an identity morphisms

$(f,g) \;\colon\; (c_1, d_1) \overset{(id_{c_1},g)}{\longrightarrow} (c_1,d_2) \overset{(f,id_{c_2})}{\longrightarrow} (c_2,d_2) \,.$

Now since the pushout product (with respect to tensor product) with the initial morphism $(\ast \to c_1)$ is equivalently the tensor product

$(\ast \to c_1) \Box_{\otimes} g \;\simeq\; id_{c_1} \otimes g$

and

$f \Box_{\otimes} (\ast \to c_2) \;\simeq\; f \otimes id_{c_2}$

the pushout-product axiom (def. ) implies that on the subcategory of cofibrant objects the functor $\otimes$ preserves acyclic cofibrations. (This is why one speaks of a Quillen bifunctor, see also Hovey 99, prop. 4.3.1).

Hence $\otimes^L$ exists.

By the same decomposition and using the universal property of the localization of a category (def.) one finds that for $\mathcal{C}$ and $\mathcal{D}$ any two categories with weak equivalences (def.) then the localization of their product category is the product category of their localizations:

$(\mathcal{C} \times \mathcal{D})[(W_{\mathcal{C}} \sqcup W_{\mathcal{D}})^{-1}] \simeq (\mathcal{C}[W^{-1}_{\mathcal{C}}]) \times (\mathcal{D}[W^{-1}_{\mathcal{D}}]) \,.$

With this, the universal property as a localization (def.) of the homotopy category of a model category (thm.) induces associators: Let

$\array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow &\swArrow_{\eta}& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{((-)\otimes^L(-))\otimes^L (-)}{\longrightarrow}& Ho(\mathcal{C}) } \;\;\;\,,\;\;\;\;\; \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{ (-) \otimes ( (-) \otimes (-) ) }{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow &\swArrow_{\eta'}& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{ (-) \otimes^L ( (-) \otimes^L (-) ) }{\longrightarrow}& Ho(\mathcal{C}) }$

be the natural isomorphism exhibiting the derived functors of the two possible tensor products of three objects, as shown at the top. By pasting the second with the associator natural isomorphism of $\mathcal{C}$ we obtain another such factorization for the first, as shown on the left below,

$(\star) \;\;\;\;\;\; \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{=}}\downarrow &\swArrow_{\alpha}& \downarrow^{\mathrlap{=}} \\ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{ (-) \otimes ( (-) \otimes (-) ) }{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow &\swArrow_\eta& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{ (-) \otimes^L ( (-) \otimes^L (-) ) }{\longrightarrow}& Ho(\mathcal{C}) } \;\;\;\;\;\; \simeq \;\;\;\;\;\; \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow &\swArrow_{\eta'}& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{((-)\otimes^L(-))\otimes^L (-)}{\longrightarrow}& Ho(\mathcal{C}) \\ {}^{\mathllap{=}}\downarrow &\swArrow_{\alpha^L}& \downarrow^{\mathrlap{=}} \\ Ho(\mathcal{C})\times Ho(\mathcal{C})\times Ho(\mathcal{C}) &\underset{(-)\otimes^L((-)\otimes^L (-))}{\longrightarrow}& Ho(\mathcal{C}) }$

and hence by the universal property of the factorization through the derived functor, there exists a unique natural isomorphism $\alpha^L$ such as to make this composite of natural isomorphisms equal to the one shown on the right. Hence the pentagon identity satisfied by $\alpha$ implies a pentagon identity for $\alpha^L$, and so $\alpha^L$ is an associator for $\otimes^L$.

The above equation on pasting composites of natural isomorphism is equivalently just the coherence law for a monoidal functor:

$\array{ (\gamma(X) \otimes^L \gamma(Y)) \otimes^L \gamma(Z) &\overset{\alpha^L_{\gamma(X), \gamma(Y), \gamma(Z)}}{\longrightarrow}& \gamma(X) \otimes^L (\gamma(Y) \otimes^L \gamma(Z)) \\ {}^{\mathllap{\eta'}}\uparrow && \uparrow^{\mathrlap{\eta}} \\ \gamma( (X \otimes Y) \otimes Z ) &\underset{\gamma(\alpha)}{\longrightarrow}& \gamma(X \otimes (Y \otimes Z)) } \,.$

## Examples

### Simplicial presheaves

If the underlying site has finite products, then both the injective and the projective, the global and the local model structure on simplicial presheaves becomes a cartesian monoidal model category with respect to the standard closed monoidal structure on presheaves.

See at model structure on simplicial presheaves the section Closed monoidal structure.

### Monoidal presheaves

More generally, if a small category $\mathcal{S}$ has finite products and $\mathcal{M}$ is a cofibrantly generated symmetric monoidal model category, then the functor category $Func\big(\mathcal{S}^{op}, \mathcal{M}\big)$ with its object-wise monoidal category-structure and with the projective model structure on functors is itself a monoidal model category (Pavlov & Scholbach 2018, inside proof of Prop. 7.9).

### Homological algebra and stable homotopy theory

With respect to a symmetric monoidal smash product of spectra:

The standard example of a monoidal model category whose unit is not cofibrant is the category of EKMM S-modules.

### Model structure on $G$-objects

###### Assumption

Let $\mathcal{E}$ be a category equipped with the structure of

such that

• the model structure is cofibrantly generated;

• the tensor unit $I$ is cofibrant.

###### Proposition

Under these conditions there is for each finite group $G$ the structure of a monoidal model category on the Borel model structure $\mathcal{E}^{\mathbf{B}G}$ of objects in $\mathcal{E}$ equipped with a $G$-action, for which the forgetful functor

$\mathcal{E}^{\mathbf{B}G} \to \mathcal{E}$

preserves and reflects fibrations and weak equivalences.

See for instance (BergerMoerdijk 2.5).

### Model structure on monoids

The first mention of monoidal model categories (without the unit axiom) under this name is in

Although the earliest mentions of terminology appear to be the sources indicated above, the notion itself is older. In particular, in his book Hovey credits the definition of a Quillen bifunctor to Dwyer–Hirschhorn–Kan–Smith, and this definition is itself based on the axiom SM7 in Quillen’s Homotopical Algebra.

The unit axiom together with the fact that the homotopy category is monoidal in this case is due to Hovey.

Textbook accounts:

Some relevant homotopy category background is in

The monoidal structures for a symmetric monoidal smash product of spectra are due to

Further variation of the axiomatics is discussed in

The monoidal model structure on the Borel model structure $\mathcal{E}^{\mathbf{B}G}$ is discussed for instance in

Relation to symmetric monoidal (infinity,1)-categories (in particular, that every locally presentable symmetric monoidal $(\infty,1)$-category arises from a symmetric monoidal model category) is discussed in

Monoidal Reedy model structures are discussed in

Monoidal presheaf categories make an appearance in